Spectrum and Thermodynamics of the One-Dimensional Supersymmetric t-J Model with 1/r 2 Exchange and Hopping

Abstract

We derive the spectrum and thermodynamics of the 1D supersymmetric t-J model with long range hopping and spin exchange using a set of maximal spin eigenstates.This spectrum confirms the recent conjecture that the asymptotic Bethe-ansatz spectrum is exact. By empirically determining the spinon degeneracies of each state, we are able to explicitly construct the free energy. number: 71.30.+h, 74.65+n, 75.10.Jm Typeset Using REVTEX PACS 1The explicit construction of low dimensional models with Jastrow ground-state wavefunctions has attracted considerable recent interest [1][2][3][4][5][6]. In one dimension, Shastry and Haldane [5,6] have demonstrated that the ground-state of the 1D Heisenberg model with a 1/r 2 exchange interaction is a Gutzwiller state for the half filled infinite-U Hubbard model.Haldane has shown how the spectrum of this model can be written in terms of a generalized type of Jastrow wavefunction with excitations of novel statistics [7]. Kuramoto and Yokoyama [8] have recently extended these results to include holes, demonstrating that the corresponding 1D supersymmetric t-J model is also characterized by a Gutzwiller ground-state. Most recently, Kawakami has obtained an asymptotic Betheansatz (ABA) solution for the model, based on the observation that the ground-state wavefunction is a product of two-body functions [9]. Assuming factorizability, he derived the spectrum of the system, which was conjectured to be exact. The low-energy critical behavior of the model has been identified as a Luttinger liquid [10,11]; the spin and charge excitations are described independently by c = 1 conformal field theories.In the case of the 1/r 2 Bose gas [12], and the Shastry-Haldane 1/r 2 Heisenberg chain [7], the ABA has been shown to furnish the correct spectrum, despite the long-range nature of the interactions. A remarkable feature of these models is that excited states are obtained from the ground-state by introducing zeros into the Jastrow wavefunction, in a manner reminiscent of Laughlin's description of quasiparticles in the fractional quantum Hall effect.This motivates us to examine the 1/r 2 supersymmetric t-J model in a similar vein. Here, we show how this philosophy can be used to construct the excited state Jastrow wavefunctions of the 1/r 2 supersymmetric t-J model and indeed, the spectrum confirms Kawakami's conjecture. In addition to the spectrum, we are able to obtain the spin degeneracies of each state, permitting us to write the the free energy in closed form.The Hamiltonian for the one-dimensional t-J model is given bywhere we implicitly project out any double occupancies. We take t ij = J ij = t/d 2 (i−j) where

Supersymmetric t-J models with long-range interactions: partition function and spectrum

“…However, this method does not completely determine the spectrum, since it does not provide complete information on the degeneracy of each level. Based on numerical calculations, Wang et al [17] proposed an empirical description of the degeneracies of the spectrum of the su (2) KY model reminiscent of the rule for filling the border strips associated to Haldane's motifs [18][19][20]. This description, however, is known to be incorrect in certain situations, although the needed corrections vanish in the thermodynamic limit [17].…”

“…Based on numerical calculations, Wang et al [17] proposed an empirical description of the degeneracies of the spectrum of the su (2) KY model reminiscent of the rule for filling the border strips associated to Haldane's motifs [18][19][20]. This description, however, is known to be incorrect in certain situations, although the needed corrections vanish in the thermodynamic limit [17]. Inspired by the equivalence between the su (2) KY model and the su(1|2) supersymmetric HS chain (up to a term proportional to the total electric charge), Saiga and Kuramoto [21] conjectured a description of the former model's spectrum essentially in terms of su(1|2)-supersymmetric Haldane motifs, which accounted for the numerical results for N 16 spins.…”

Local Hamiltonians for one-dimensional critical models

“…It is likely that the excited states of the one-dimensional models for other values of q are also interesting and related to the anyons that can be obtained in the 2D versions of the models. One-dimensional quantum many-body models with nonlocal interactions of the kind 1/r 2 , as well as truncated versions of them, have gained considerable interest over the years [3][4][5][6][7][8][9][10][11][12][13][14][15], exploring interesting aspects of integrable systems. A common feature of quant um 1/r 2 systems is that the ground state can be exactly represented by a Jastrowtype wavefunction [3][4][5][6][7][8][9][10][11][12][13][14][15], namely a product of two-body functions.…”